Hurwitz’s theorem

Define the ball at z0 of radius r as B(z0,r)={zG:|z-z0|<r} and D(z0,r)={zG:|z-z0|r} is the closed ballPlanetmathPlanetmath at z0 of radius r.

Theorem (Hurwitz).

Let GC be a region and suppose the sequence of holomorphic functionsMathworldPlanetmath {fn} converges uniformly on compact subsets of G to a holomorphic function f. If f is not identically zero, D(z0,r)G and f(z)0 for z such that |z-z0|=r, then there exists an N such that for all nN f and fn have the same number of zeros in B(z0,r).

What this theorem says is that if you have a sequence of holomorphic functions which converge uniformly on compact subsets (such a sequence always convergesPlanetmathPlanetmath to a holomorphic function but that’s another theorem altogether), the function is not identically zero and furthermore the function is not zero on the boundary of some ball, then eventually the functions of the sequence have the same number of zeros inside this ball as does the function.

Do note the requirement for f not being identically zero. For example the sequence fn(z):=1n converges uniformly on compact subsets to f(z):=0, but fn have no zeros anywhere, while f is identically zero.

Also in general this result holds for boundedPlanetmathPlanetmathPlanetmathPlanetmath convex subsets ( but it is most useful to for balls.

An immediate consequence of this theorem is this useful corollary.


If G is a region and a sequence of holomorphic functions {fn} converges uniformly on compact subsets of G to a holomorphic function f, and furthermore if fn never vanishes (is not zero for any point in G), then f is either identically zero or also never vanishes.


  • 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
Title Hurwitz’s theorem
Canonical name HurwitzsTheorem
Date of creation 2013-03-22 14:17:55
Last modified on 2013-03-22 14:17:55
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 5
Author jirka (4157)
Entry type Theorem
Classification msc 30C15
Related topic CompositionAlgebraOverAlgebaicallyClosedFields
Related topic CompositionAlgebrasOverMathbbR
Related topic CompositionAlgebrasOverFiniteFields
Related topic CompositionAlgebrasOverMathbbQ